Eee 188 Review Problems Date 05012018 Problem 1 Controllabil

1eee 188 Review Problemsdate 05012018problem 1 Controllability Of

Review Problems from the 2018 course; specifically, problem 1 regarding the controllability of a two-tank hydraulic system, along with related control design tasks. The assignment involves analyzing system controllability, designing controllers using Ziegler-Nichols tuning, state feedback with integral action, observer design, and control of a cruise control system, as well as controllability and observability of a coupled tank system, and PI controller design for liquid level regulation.

Paper For Above instruction

The problems presented involve advanced control system analysis and design in continuous and discrete domains, emphasizing the importance of controllability, observability, and effective controller synthesis using state feedback, integral action, and pole placement techniques.

Controllability of a Two-Tank System

The two-tank hydraulic system illustrated in figure 1 presents a classic example of multilayer control analysis. Without performing detailed calculations, the question is whether the system is controllable purely based on its description. Since the input cannot influence the first tank's level directly, but the levels are coupled through flow between tanks, the controllability depends on the interconnection and the inputs affecting the system state variables.

In general, if the control inputs can influence both tank levels, directly or indirectly, the system may be controllable. However, the key point here is whether the input can change the states independently. Given the physical configuration, it is plausible that the system is controllable because the flow between tanks results from the input and internal dynamics, allowing control over the levels via proper control signals.

Assuming controllability, the subsequent tasks involve designing a PID controller based on Ziegler-Nichols tuning, which relies on experimental determination of ultimate gain and period. Additionally, a state feedback controller with integral action can be designed to place the closed-loop poles at specified locations (0.15, 0.11, 0.55), ensuring desired transient response and zero steady-state error in presence of disturbances.

Observability without the Output Matrix

Consider the system described by the scalar difference equation:

x(k + 1) = -0.25x(k) + αu(k)

To analyze observability without explicit output matrix information, observe that observability depends on whether the initial state can be reconstructed from the output measurements. Since the system is scalar, the key factor is whether the system's dynamics are detectable given the available output.

In this case, the value of α affects the influence of the input on the system's state evolution but does not directly impact observability related to the state equation. For a scalar system, observability is always maintained provided the output Y(k) is directly measuring the state x(k). Without the output matrix explicitly defined, if the output is the state itself, the system is always observable.

For observer design, a Luenberger observer or a Kalman filter could be utilized, assuming an appropriate output measurement. The stability of the observer depends on the eigenvalues of the system matrix and the placement of observer poles.

State Feedback for Cruise Control

The cruise control system modeled in discrete-time form is given by:

v(k + 1) = 0.95v(k) + 10^(-3)u(k)

The control goal is to maintain a constant speed of 40 mph. The augmented system includes an integral of the error to eliminate steady-state error. The augmented state vector is [v(k); z(k)] where z(k) is the integral of the velocity error.

The output matrix C, representing measurable outputs, is given by y(k) = v(k), implying that the measurement directly reflects the velocity. Thus, the output matrix is:

C = [1 0]

The state feedback gain matrix K will have dimension 2x2, corresponding to the two states v and z. Designing the state feedback to place the eigenvalues at 0.3 and 0.6 involves calculating the feedback gains (K1, K2) that modify the system's A matrix to achieve desired pole locations via pole placement methods such as Ackermann’s formula.

Eigenvalue placement ensures the system's response converges quickly to the reference speed without oscillations or sluggish behavior. The eigenvalues’ proximity to the unit circle's interior influences the transient response and stability margins.

The numerical value of the closed-loop system matrix (A - BK) can be computed once K is determined. Its eigenvalues will match the desired placements, confirming the controller’s effectiveness.

The second variable in the augmented system (variable z) models the integral of the error, which helps eliminate steady-state error by adjusting the control input proportionally to accumulated error over time.

Controllability and Observability of a Coupled Tank System

In two coupled tanks with fluid levels h1 and h2, the continuous dynamics are described by differential equations reflecting inflows and outflows. Discretizing with T=1s yields the state-space equations with states h1(k), h2(k). The input vector includes the flow Qi and control variables, while qb represents a disturbance flow between tanks, which cannot be controlled directly.

Constructing matrices A, B, and C involves linearizing the system equations and formulating the discrete-time model. With the assumption that qb=0 for controllability and observability analysis, we can verify whether the pair (A, B) and (A, C) are of full rank.

Calculating the controllability matrix involves stacking the matrices [B, AB, A^2B], while the observability matrix encompasses [C, CA, CA^2]. If these matrices are of full rank (equal to the system's dimension), the system is controllable and observable, respectively. Conducting rank tests confirms the properties—likely, the two-tank system is controllable and observable under standard assumptions.

Errors e1(k) and e2(k) can be defined as the differences between the actual tank levels and their reference values, providing a basis for feedback control law design aimed at minimizing these errors.

PI Control Design for Liquid Level Regulation

Designing PI controllers involves tuning proportional gains and integral gains to achieve desired stability and response characteristics. The transfer functions for the tanks include included respective pole placements—placing them at 0.5 for Tank 2 and at 0.5 and 0.4 for Tank 1—via pole placement techniques such as Ackermann’s formula.

The stability interval for the proportional gain K3 on Tank 2 can be derived from the characteristic equation, ensuring the poles stay within the unit circle for discrete-time stability. Similarly, tuning K1, K2, and K3 involves solving algebraic equations for pole placement, possibly via Ackermann's formula, to achieve specified response times and zero steady-state error.

The steady-state error assessments involve analyzing the system’s response to constant inputs. Zero steady-state error generally requires integral action or appropriate controller design—especially pertinent for Tank 2, where proportional control alone cannot eliminate the steady-state error due to disturbance or model uncertainties.

At k=7, control actions for qi and qc are calculated based on errors and their integrals, with specific gains and initial conditions provided. These computations confirm the effectiveness of the control strategy and help refine controller gains further if necessary.

Conclusion

These problems collectively emphasize understanding system properties like controllability and observability, alongside practical controller design techniques such as PID tuning and pole placement. Properly designed controllers improve system stability, transient response, and disturbance rejection, essential in hydraulic and vehicle control applications.

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